Hitting Times, Cover Cost, and the Wiener Index of a Tree

TL;DR

This paper explores the relationship between hitting times of random walks, the Wiener index, and graph invariants, revealing a simple identity for trees and analyzing vertex preorders in graphs.

Contribution

It establishes a new identity linking hitting times and the Wiener index specifically for trees and studies vertex preorders in graphs.

Findings

Hitting times relate directly to the Wiener index in trees.

Vertices can be ordered by ease of reach and exit in random walks.

Preorders coincide in trees but differ in general graphs.

Abstract

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are "easier to reach" by a random walk, but "more difficult to get out of". We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behaviour or random walk on a graph to its eigenvalues in a new perspective.